Abstract: This paper addresses the problem of averaging numbers across a wirelessnetwork from an important, but largely neglected, viewpoint: bandwidth-energyefficiency. We show that existing distributed averaging schemes have severaldrawbacks and are inefficient, producing networked dynamical systems thatevolve with wasteful communications. Motivated by this, we develop ControlledHopwise Averaging CHA, a distributed asynchronous algorithm that attempts to-make the most- out of each iteration by fully exploiting the broadcast natureof wireless medium and enabling control of when to initiate an iteration. Weshow that CHA admits a common quadratic Lyapunov function for analysis, derivebounds on its exponential convergence rate, and show that they outperform theconvergence rate of Pairwise Averaging for some common graphs. We alsointroduce a new way to apply Lyapunov stability theory, using the Lyapunovfunction to perform greedy, decentralized, feedback iteration control. Finally,through extensive simulation on random geometric graphs, we show that CHA issubstantially more efficient than several existing schemes, requiring far fewertransmissions to complete an averaging task.